Abstract

We consider the following mixed type cubic and quartic functional equation = where is a fixed integer. We establish the general solution of the functional equation when the integer , and then, by using the fixed point alternative, we investigate the generalized Hyers-Ulam-Rassias stability for this functional equation when the integer .

1. Introduction

In 1940, Ulam [1] asked the fundamental question for the stability for the group homomorphisms.

Let be a group, and let be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality
for all , then there is a homomorphism with
for all ?

In other words, under what conditions, does there exist a homomorphism near an approximately homomorphism? In the next year, Hyers [2] gave the first affirmative answer to the question of Ulam for Cauchy equation in the Banach spaces. Then, Rassias [3] generalized Hyers’ result by considering an unbounded Cauchy difference, and this stability phenomenon is known as generalized Hyers-Ulam-Rassias stability or Hyers-Ulam-Rassias stability. During the last three decades, the stability problem for several functional equations has been extensively investigated by many mathematicians; see, for example, [4–9] and the references therein. We also refer the readers to the books [10–13].

In [14], Jun and Kim introduced the following functional equation
It is easy to see that the function satisfies the functional equation (3). Thus, it is natural that (3) is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic function. In [14], Jun and Kim established the general solution and the generalized Hyers-Ulam-Rassias stability for (3). They proved that a function between real vector spaces is a solution of the functional equation (3) if and only if there exists a function such that for all and is symmetric for each fixed one variable and is additive for fixed two variables.

In [15], Lee et al. considered the following quartic functional equation
Since the function satisfies the functional equation (4), the functional equation (4) is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function. In [15], the authors solved the functional equation (4) and proved the stability for it. Actually, they obtained that a function between real vector spaces satisfies the functional equation (4) if and only if there exists a symmetric biquadratic function such that for all . A function between real vector spaces is said to be quadratic if it satisfies the following functional equation
for all , and a function is said to be bi-quadratic if is quadratic for each fixed one variable (see [8]). The following mixed type cubic and quartic functional equation was introduced by Eshaghi Gordji et al. [16]
It can be verified that the function satisfies the functional equation (6). In [16], the authors obtained the general solution and the generalized Hyers-Ulam-Rassias stability of (6) in quasi-Banach space. The literature on the stability of the mixed type functional equations is very rich; see [17–22].

In the present paper, we extend (6) and consider the following functional equation
where is a fixed integer. One can see that the functional equation (6) is a special case of (7) when we take the integer .

In 2003, Radu [23] noticed that the fixed point theorem, which was established by Diaz and Margolis [24], plays an important part in solving the stability problem of functional equations. Subsequently, this method has been successfully used by many mathematicians to investigate the stability of several functional equations; see, for example, [21, 25, 26] and the references therein.

In this paper, we first establish the general solution of functional equation (7) when the integer and is a mapping between vector spaces. Then, by using the fixed point method, we prove the generalized Hyers-Ulam-Rassias stability of the functional equation (7) when the integer and is a mapping from the normed space to the Banach space.

Recall form [14, 15] that every solution of the cubic functional equation (3) and the quartic functional equation (4) is said to be a cubic function and a quartic function, respectively. In this section, we investigate the general solution of the mixed type cubic and quartic functional equation (7). Throughout this section, let and be two real vector spaces, and we always assume that the integer in the functional equation (7) is different from 0, −1, and 1. Before proving our main theorem, we first give the following two lemmas.

Proof. Note that, in view of the oddness of , we have and for all . Letting in (7), we get
for all . Applying (8) to (7), we obtain
for all . Replacing by in (9), we get
for all . Now, if we replace by in (9) and use (8), we see that
for all . Interchanging with in (11) and using the oddness of , we get the relation
for all . Then, subtracting (12) from (10), one has
for all . Now, replacing by in (13) and noting that is odd, we have
for all . Adding (13) to (14) gives
for all . Replacing by in (9) and using (8), we get
Applying (9) and (16) to (15) yields that
for all . Thus, the mapping is cubic. This completes the proof.

Proof. In view of the evenness of , we have for all . Putting in (7), we get . Then, let in (7), we obtain
for all . Combing (7) and (18) implies the following equation
for all . Replacing by in (19) and note that is even, we get
for all . Replacing by in (19) and using (18), we get
for all . Interchanging the roles of and in (21), we obtain
for all . If we subtract (22) from (20), we obtain
for all . Replacing by in (23), we get
for all . If we add (23) to (24), we have
for all . Replacing by in (19) and using (18), we get
for all . Applying (19) and (26) to (25), we obtain that
for all . Therefore, the mapping is quartic and the proof is complete.

Theorem 3. A mapping satisfies (7) for all if and only if there exist a symmetric multiadditive mapping and a symmetric bi-quadratic mapping such that for all .

Proof. First, we assume that there exist a symmetric multi-additive mapping and a symmetric bi-quadratic mapping such that for all . We need to show that the function satisfies (7). By a simple computation, one can obtain that the function satisfies (7). Thus, to show that the function satisfies (7), we only need to show that the function also satisfies (7); namely,
for all . Since is a bi-quadratic mapping, it can be verified that
for all integer and all . Then, (28) becomes
To establish (28), it suffices to show (30). Note that if (30) holds for some integer , then so does . Thus, in the following, we will show that (30) holds for all positive integers with and all . To do this, we use induction on . Fix any . In the case when , we have
So (30) is true for . Here, we have used the bi-quadratic property of the function and (29). Now, assume that (30) is true for all positive integers that are less than or equal to some integer . Then,
Applying (31) to (32), one can obtain that
This means that (30) is true for , and we have showed that the function satisfies (7). Therefore, the mapping satisfies (7).Conversely, we decompose into the odd part and the even part by putting
for all . Then, for all . It is easy to show that the mappings and satisfy (7). Hence, it follows from Lemmas 1 and 2 that the function is cubic and is quartic, respectively. Therefore, there exist a symmetric multi-additive mapping such that for all (see [14, Theorem 2.1]) and a symmetric bi-quadratic mapping such that for all (see [15, Theorem 2.1]). Hence, we get for all . The proof is complete.

In this section, we will investigate the stability of the functional equation (7) by using the fixed point alternative. Throughout this section, let be a real normed space and be a real Banach space, and we always assume that the integer used in the section is greater than or equal to 2. For convenience, we use the following abbreviation for a given function
for all .

Let us recall the following result by Diaz and Margolis.

Proposition 4 (see [24]). Let be a complete generalized metric space (i.e., one for which may assume infinite value), and let be a strictly contractive mapping with Lipschitz constant ; that is,
Then, for each fixed element , either
for all nonnegative integers or there exists a non-negative integer such that(a) for all ;(b)the sequence converges to a fixed point of ;(c) is the unique fixed point of in the set ;(d) for all .

Lemma 5. Let be an odd function for which there exists a function such that
for all . If there exists a constant such that
for all , then there exists a unique cubic mapping such that
for all , where .

Proof. It follows from (39) that
for all . Letting in (38) and replacing by , we have
for all . By (39), we have for all . This, together with (42), implies that
for all . Let be the set of all odd mappings . We introduce the generalized metric on :
It is easy to show that is complete. Now, we define a function by
for all and all . Note that, for all ,
Hence, we obtain that
for all ; that is, is a strictly contractive mapping of with Lipschitz constant . It follows from (43) that . Therefore, according to Proposition 4, the sequence converges to a fixed point of ; that is,
and for all . Also is the unique fixed point of in the set and
which yields the inequality (40). It follows from the definition of , (38), and (41) that
for all ; that is, the mapping satisfies (7). Since is odd, is odd. Therefore, Lemma 1 guarantees that is cubic. Finally, it remains to prove the uniqueness of . Let be another cubic function satisfying (40). Since and is cubic, we get and for all ; that is, is a fixed point of . Since is the unique fixed point of in , it follows that .

Lemma 6. Let be an odd function for which there exists a function such that
for all . If there exists a constant such that
for all , then there exists a unique cubic mapping such that
for all , where .

Proof. It follows from (52) that
for all . Letting in (51) and replacing by , we have
for all . We introduce the same definitions for and as in the proof of Lemma 5 (by replacing by ) such that becomes a generalized complete metric space. Let be the mapping defined by
for all and all . One can show that for all . It follows from (55) that . Due to Proposition 4, the sequence converges to a fixed point of ; that is,
and for all . Also,
which yields the inequality (53). The rest of the proof is similar to the proof of Lemma 5, and we omit the details.

Similarly, we can prove the following two lemmas on even functions.

Lemma 7. Let be an even function with for which there exists a function such that
for all . If there exists a constant such that
for all , then there exists a unique quartic mapping such that
for all , where .

Lemma 8. Let be an even function with for which there exists a function such that
for all . If there exists a constant such that
for all , then there exists a unique quartic mapping such that
for all , where .

Now, we are ready to give our main theorems in this section.

Theorem 9. Let be a function with for which there exists a function such that
for all . If there exists a constant such that
for all , then there exist a unique cubic mapping and a unique quartic mapping such that
for all , where

Proof. Let and denote the odd and the even part of , respectively. Then, it can be verified from (65) that
for all . Moreover, by (66), it is easy to compute that
for all . Thus, by applying Lemmas 5 and 7, one can obtain that there exist a unique cubic mapping and a unique quartic mapping such that
for all , where and . Moreover, combining (71) and (72) yields the inequality (67). The proof is complete.

Theorem 10. Let be a function with for which there exists a function such that
for all . If there exists a constant such that
for all , then there exist a unique cubic mapping and a unique quartic mapping such that
for all , where

Proof. Similar to the proof of Theorem 9, the result follows from Lemmas 6 and 8.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.